A note on sum-product estimates over finite valuation rings
Volume 198 / 2021
Abstract
Let $\mathcal R $ be a finite valuation ring of order $q^r$ with $q$ a power of an odd prime number, and let $\mathcal A \subset \mathcal R $. We improve a recent result due to Yazici (2018) on a sum-product type problem. More precisely, we prove that
$\bullet$ if $|\mathcal A |\gg q^{r- {1}/{3}}$, then \[\max \lbrace |\mathcal A +\mathcal A |, |\mathcal A ^2+\mathcal A ^2| \rbrace \gg q^{ {r}/{2}}|\mathcal A |^{ {1}/{2}};\]
$\bullet$ if $q^{r- {3}/{8}}\ll |\mathcal A |\ll q^{r- {1}/{3}}$, then \[\max \lbrace |\mathcal A +\mathcal A |, |\mathcal A ^2+\mathcal A ^2| \rbrace \gg \frac {|\mathcal A |^2}{q^{ {(2r-1)}/{2}}};\]
$\bullet$ if $|\mathcal A +\mathcal A |\,|\mathcal A |^2\gg q^{3r-1}$ and $2q^{r-1}\le |\mathcal A |\ll q^{r- {3}/{8}}$, then \[\max \lbrace |\mathcal A +\mathcal A |, |\mathcal A ^2+\mathcal A ^2| \rbrace \gg q^{r/3}|\mathcal A |^{2/3}.\]
